Abstract

We investigate several results concerning the differential subordination of analytic and multivalent functions which is defined by using a certain fractional derivative operator. Some special cases are also considered.

1. Introduction and Definitions

Let denote the class of functions of the form

which are analytic in the open unit disk Also let denote the class of all analytic functions with which are defined on . If and are analytic in with , then we say that is said to be subordinate to in , written or , if there exists the Schwarz function , analytic in such that , and In particular, if the function is univalent, then the above subordination is equivalent to and .

Let , and be complex numbers with Then the Gaussian hypergeometric function is defined by

where is the Pochhammer symbol defined, in terms of the Gamma function, by

The hypergeometric function is analytic in and if or is a negative integer, then it reduces to a polynomial.

There are a number of definitions for fractional calculus operators in the literature (cf., e.g., [1, 2]). We use here the Saigo-type fractional derivative operator defined as follows (see [3]; see also [4]).

Definition 1.1. Let and . Then the generalized fractional derivative operator of a function is defined by
The function is an analytic function in a simply-connected region of the -plane containing the origin, with the order
for and the multiplicity of is removed by requiring that be real when .

Definition 1.2. Under the hypotheses of Definition 1.1, the fractional derivative operator of a function is defined by

With the aid of the above definitions, we define a modification of the fractional derivative operator by

Note that , and , where is the fractional derivative operator defined by Srivastava and Aouf [5, 6].

In this manuscript, we will use the method of differential subordination to derive certain properties of multivalent functions defined by fractional derivative operator .

2. Main Results

In order to establish our results, we require the following lemma due to Miller and Mocanu [7].

Lemma 2.1. Let be univalent in and let and be analytic in a domain containing with when . Set , and suppose that(1) is starlike (univalent) in ,(2).If is analytic in , with , , and
then and is the best dominant.

We begin by proving the following

Theorem 2.2. Let and , and let , , , and . Suppose that is univalent in and satisfies
If and
then
and is the best dominant.

Proof. Define the function by
Then is analytic in with . A simple computation using (2.5) gives
By applying the identity (1.9) in (2.6), we obtain
Making use of (2.5) and (2.7), we have
In view of (2.8), the subordination (2.3) becomes
and this can be written as (2.1), where
Since , we find from (2.10) that and are analytic in with . Let the functions and be defined by
Then, by virtue of (2.2), we see that is starlike and
Hence, by using Lemma 2.1, we conclude that , which completes the proof of Theorem 2.2.

Remark 2.3. If we put in Theorem 2.2, then we get new subordination result for the fractional derivative operator due to Srivastava and Aouf [5, 6].

Theorem 2.4. Let and , and let , , , and . Suppose that is univalent in and satisfies
If and
then
and is the best dominant.

Proof. Define the function by
Then is analytic in with . By a simple computation, we find from (2.16) that
By using the identity (1.9) in (2.17), we obtain
Applying (2.16) and (2.18), we have
In view of (2.19), the subordination (2.14) becomes
and this can be written as (2.1), where
Since , it follows from (2.21) that and are analytic in with . Let the functions and be defined by
Then, by virtue of (2.13), we see that is starlike and
Hence, by using Lemma 2.1, we conclude that , which proves Theorem 2.4.

Theorem 2.7. Let and , and let , , , and . Suppose that is univalent in and satisfies
If and
then
and is the best dominant.

Proof. Define the function by
Then is analytic in with . A simple computation using (1.9) and (2.31) gives
By using (2.29), (2.31), and (2.32), we get
And this can be written as (2.1) when and . Note that and and are analytic in . Let the functions and be defined by
Then, by virtue of (2.28), we see that is starlike and
Hence, by applying Lemma 2.1, we observe that , which evidently proves Theorem 2.7.

Finally, we prove

Theorem 2.8. Let and , and let , , , and . Suppose that be univalent in and satisfies
If and
then
and is the best dominant.

Proof. If we define the function by
then is analytic in with . Hence, by using the same techniques as detailed in the proof of Theorem 2.2, we obtain the desired result.

By taking in Theorem 2.8 and after a suitable change in the parameters, we have the following.

Corollary 2.9. Let and . Suppose that is univalent in and satisfies
If and
then and is the best dominant.

Acknowledgment

This work was supported by Daegu National University of Education Research Grant in 2008.